IIT JEE pyq with solutions

VECTOR ALGEBRA


VECTOR ALGEBRA
  
1. Let aˆ, bˆ be unit vectors. If c be a vector such that the 6. Let a  iˆ  ˆj  kˆ and c  2iˆ  3iˆ  2k . Then the
   
    number of vectors b such that b  c  a and
angle between â and c is , and bˆ  c  2  c  aˆ  , 
12
b  1, 2,........,10 is :
2
then 6c is equal to
(a) 0 (b) 1
(a) 6 3  3  (b) 3  3 (c) 2
 
(d) 3
7. Let a and b be the vectors along the diagonal of a
(c) 6 3  3  (d) 6  3 1  parallelogram having area 2 2. let the angle between
2. Let â and b̂ be two unit vectors such that       
a and b be acute. a  1 and a  b  a  b . If
  
aˆ  bˆ  2 aˆ  bˆ  2.  If    0,   is the angle    

 
c  2 2 a  b  2b , then the angle between b and c

between â and bˆ, then among the statements :
is :
 S1 : 2 aˆ  bˆ  aˆ  bˆ  
(a) (b) 
4 4
 S 2  : The projection of 
â on â  bˆ is 1
2 (c)
5
(d)
3
(a) Only (S1) is true 6 4
 
(b) Only (S2) is true 8. Let a   iˆ  2 ˆj  kˆ and b  2iˆ   ˆj  kˆ, where
(c) Both (S1) and (S2) are true   R. If the area of the parallelogram whose adjacent
(d) Both (S1) and (S2) are false  
 sides are represented by the vectors a and b is
3. Let a  a1iˆ  a2 ˆj  a3 kˆ ai  0, i  1, 2,3 be a vector 2   2
which makes equal angles with the coordinates axes
15  2  4  , then the value of 2 a  a  b b is  

OX , OY and OZ . Also, let the projection of a on the equal to
 (a) 10 (b) 7
vector 3iˆ  4 ˆj be 7. Let b be a vector obtained by
   (c) 9 (d) 14
rotating a with 90 . If a , b and x-axis are coplanar, 9.

Let a be a vector which is perpendicular to the vector

then projection of a vector b on 3iˆ  4 ˆj is equal to
3iˆ 
1ˆ
2

 
j  2kˆ. If a  2iˆ  kˆ  2iˆ  13 ˆj  4kˆ, then the
(a) 7 (b) 2

(c) 2 (d) 7 projection of the vector a on the vector 2iˆ  2 ˆj  kˆ is
      
4. If a  b  1, b  1, b  c  2 and c  a  3, then the value 1
(a) (b) 1
         3
  
of  a  b  c , b   c  a  , c  b  a  is :
   5 7
   (c) (d)
(a) 0 (b) 6a  b  c   3
 
3
      10. Let a   iˆ  3 ˆj  kˆ, b  3iˆ   ˆj  4k and
(c) 12c  a  b   (d) 12b  c  a 
 ˆ
   c  i  2 ˆj  2k ,  , R , be three vectors. If the
5. Let a  iˆ  ˆj  2kˆ, b  2iˆ  3 ˆj  kˆ are c  iˆ  ˆj  k be
    10
three given vectors. Let v be a vector in the plane of a projection of a on c is and
3
  2   
and b whose projection on c is . If v  ˆj  7, then b  c  6iˆ  10 ˆj  7 kˆ, then the value of    equal
3
to :
 ˆ ˆ
 
v  i  k is equal to (a) 3 (b) 4
(a) 6 (b) 7 (c) 5 (d) 6
(c) 8 (d) 9 11. Let A, B, C be three points whose position vectors
respectively are :

, VECTOR ALGEBRA


a  iˆ  4 ˆj  3kˆ 4 5
 (c) (d)
5 6
b  2iˆ   ˆj  4kˆ,   R 

 16. Let a   iˆ  ˆj   kˆand b  3i  5 ˆj  4kˆ be two
ˆ
c  3iˆ  2 ˆj  5kˆ
 
   vectors, such that a  b  iˆ  9iˆ  12kˆ. Then the
If  is the smallest positive integer for which a, b , c    
projection of b  2a on b  a is equal to
ar e non-collinear, then the length of the median, in
ABC , through A is : 39
(a) 2 (b)
5
82 62
(a) (b) 46
2 2 (c) 9 (d)
5
69 66  
(c) (d) 17. Let a  2iˆ  ˆj  5kˆ and b   iˆ   ˆj  2kˆ. If
2 2
  23 
12. Let ABC be
      
a triangle

such that
 
a  b  iˆ  kˆ  , then b  2 ˆj is equal to
2
BC  a , CA  b , AB  c , a  6 2, b  6 3, and
(a) 4 (b) 5
 
b  c  12 consider the statements : (c) 21 (d) 17
    
   
 
 S1 : a  b  c  b  c  6 2 2  1  
18. Let vector a has a magnitude 9. Let a vector b be
such that for every  x, y   R  R   0, 0  , the vector
 2
 S 2  : ABC  cos   . Then
1
 
 3  
  
xa  yb is perpendicular to the vector 6 ya  18 xb . 
(a) Both (S1) and (S2) are true  
Then the value of a  b is equal to :
(b) Only (S1) is true
(c) Only (S2) is true (a) 9 3 (b) 27 3
(d) Both (S1) and (S2) are false (c) 9 (d) 81
 
13. Let a  iˆ  ˆj  2kˆ and b be a vector such that 19. Let S be the set of all a  R for which the angle
     
between the vectors u  a  log e b  iˆ  6 j  3kˆ and
a  b  2iˆ  kˆ and a  b  3. Then the projection of b
  
on the vector a  b is :- v   log e b  iˆ  2 j  2a  log e b  kˆ,  b  1 is acute.
2 3 Then S is equal to:
(a) (b) 2
21 7  4
(a)  – ,   (b) 
2 7 2  3
(c) (d)
3 3 3  4   12 
 (c)   , 0  (d)  ,  

14. Let a   iˆ  ˆj  k and b  2iˆ  ˆj   kˆ, and   0. If
ˆ  3   7 
  
 
the projection of a  b on the vector i  2 ˆj  2kˆ is 20. Let a  3iˆ  ˆj and b  iˆ  2 ˆj  kˆ. Let c be a vector
     
30, then  is equal to  
satisfying a  b  c  b   c. If b c are non-parallel,
15 then the value of  is:
(a) (b) 8
2 (a) 5 (b) 5
13 (c) 1 (d) –1
(c) (d) 7
2 21. Let â and b̂ be two unit vectors such that the angle

15. A vector a is parallel to the line of intersection of the 
between them is . If  is the angle between the
plane determined by the vectors iˆ, iˆ  ˆj and the plane 4
determined by the vectors iˆ  ˆj , iˆ  kˆ. The obtuse
 
vectors  â  bˆ    
and aˆ  2bˆ  22 aˆ  bˆ , then the
angle between a and the vector b  iˆ  2 ˆj  2kˆ is value of 164 cos 2  is equal to :
3 2 (a) 90  27 2 (b) 45  18 2
(a) (b)
4 3
(c) 90  3 2 (d) 54  90 2